Multiscale Finite Element Methods for Flows on Rough Surfaces (Communications in Computational Physics, 2013)
In this paper, we present the Multiscale Finite Element Method (MsFEM)for problems on rough heterogeneous surfaces. We consider the diffusion equation onoscillatory surfaces. Our objective is to represent small-scale features of the solutionvia multiscale basis functions described on a coarse grid. This problem arises in manyapplications where processes occur on surfaces or thin layers. We present a unifiedmultiscale finite element framework that entails the use of transformations that mapthe reference surface to the deformed surface. The main ingredients of MsFEM are (1)the construction of multiscale basis functions and (2) a global coupling of these basisfunctions. For the construction of multiscale basis functions, our approach uses thetransformation of the reference surface to a deformed surface. On the deformed surface, multiscale basis functions are defined where reduced (1D) problems are solvedalong the edges of coarse-grid blocks to calculate nodal multiscale basis functions. Furthermore, these basis functions are transformed back to the reference configuration.We discuss the use of appropriate transformation operators that improve the accuracyof the method. The method has an optimal convergence if the transformed surfaceis smooth and the image of the coarse partition in the reference configuration formsa quasiuniform partition. In this paper, we consider such transformations based onharmonic coordinates (following H. Owhadi and L. Zhang [Comm. Pure and AppliedMath., LX(2007), pp. 675–723]) and discuss gridding issues in the reference configuration. Numerical results are presented where we compare the MsFEM when twotypes of deformations are used for multiscale basis construction. The first deformationemploys local information and the second deformation employs a global information.Our numerical results show that one can improve the accuracy of the simulations whena global information is used.